The method you use to move the folder is what likely broke the "share". If you drag a folder using Windows Explorer or Finder, Dropbox sees that as a single operation — a simple "change the location of this folder" request — and will maintain the shared status and share link. The folder's location is moved.
If you cut and paste the folder, Dropbox sees it as two unrelated things: first the folder is deleted (even if your computer has actually made a copy on its clipboard — but Dropbox doesn't necessarily know that) and then some time later when you hit "paste", whether moments or hours later, a folder is being created with a bunch of content in it (which your computer dumps in from its clipboard). But it's a new folder, a new copy.
The difference between cut/paste versus dragging to move folders doesn't make a lot of difference to us: different means to the same end — but it's very different from a file system's point of view.
To keep shares and links alive, either stick to dragging folders (you can open multiple Explorer/Finder windows if it helps), or use the Dropbox web interface - right-click a folder and choose "Move..." on the pop up menu.